Induction Motor Torque Estimation Accuracy Using Motor Terminal Variables Lassi Aarniovuori, Hannu Kärkkäinen, Markku Niemelä, Pia Lindh and Juha Pyrhönen Laboratory of Electric Drives Technology, Lappeenranta University of Technology, Lappeenranta, Finland
[email protected] Abstract— Information about the torque produced by an electrical motor is needed in many applications such as hybrid drivetrains and in some applications where the shaft torque cannot be measured directly e.g. because of vibrations or other physical limitations. This paper investigates the torque determination accuracy using the voltage and current time domain data obtained from motor terminals using a power analyzer. The calculated torque values are compared with the measured values obtained with a state-of-the-art torque transducer. The results are analyzed in 16 different operating points with torque values of 25%, 50%, 75% and 100% of motor rated torque and supply frequency values of 25%, 50%, 75% and 100% of rated motor frequency. Measurement results with a sinusoidal generator supply and a converter PWM supply are used in the comparison. Keywords— Adjustable speed drive; Frequency converter; Induction motor; Measurement; Pulse Width Modulation; Torque estimation.
I. INTRODUCTION The power analyzers’ computational power and sampling frequencies have increased and more complex algorithms can be processed in real time. Torque estimation is commonly used in frequency converters and the estimation accuracy has been a subject of numerous publications. When using the converter’s own relatively low cost measurement instruments the accuracy may be a concern. Usually, the converter is not measuring the line-to-line or phase voltage but, instead, it calculates the voltage values using the power switch states and measured DC-voltage. As a result, common problems in the torque estimation are the DC-voltage measurement error, the current measurement error (amplitude and phase), the stator resistance estimation error and switch on-state-resistance error. In a testing environment, using the motor terminal quantities we still have the stator resistance estimation error but the other errors are not in place. The electric motor can be part of a hybrid drive drain, integrated with an actuator (e.g. pump, fan or compressor), or integrated with a gear system and the motor torque cannot be measured directly using a torque transducer. So the question is, how accurately can the motor torque be determined by using the data provided by the power analyzer and simple algorithms with easily acquirable parameters in different operating points with sinusoidal and converter supply. A similar approach is given in [1] where stator voltage and current measurement based torque estimation is performed by using complex programmable cascaded low-pass filters and a DSP applying a recurrent neural network trained algorithm. In [2], the accuracy of torque estimate provided by
the commercial converter is compared to the measured ones both in steady and dynamical states. In [3] the instantaneous values of three phase currents and flux linkages obtained with search coils are used to estimate the motor torque. Several types of observers have been introduced to obtain the machine state for control purposes [4], [5]. Commonly in an observer approach the induction motor equivalent circuit parameters are needed to estimate the torque produced by the machine. The parameter sensitivity and measurement uncertainty propagation in the torque estimation accuracy of induction motors is investigated in [6]. II.
AIR G AP TORQUE
A starting point for torque estimation algorithms is the air gap torque that for control purposes can be defined as a cross product of the stator flux linkage y and stator current space vectors 3 (1) y × . 2 Here is the number of pole pairs and the stator flux linkage vector is an integral of the stator voltage with the effect of stator resistive voltage drop removed ∗ air-gap
y =
(
=
−
)d .
(2)
These two equations are the key elements in variable speed operation of converter-fed motors. The torque and flux linkage are controlled according to application load torque and speed to achieve an energy efficient operation of the system. Equation (1) has, however, the problem that it assumes that the whole stator current can be used in torque estimation. Actually, the stator current consists of iron loss current and exiting and torque producing current ′. Therefore (1) should be rewritten in the form
3 (3) y × ′. 2 Especially, inclusion of the iron loss component in the torque evaluation causes a clear error in the torque estimation and should somehow be taken into account in the real torque evaluation. However, estimating ′ is not necessarily straightforward if the motor equivalent circuit parameters are not accurately known. Also, there remains a problem that ′ still contains the excess loss component that does not produce air-gap
=
Fig. 1. Induction motor current spectra up to 20 th harmonics with 25 Hz converter supply and with four load values.
torque. In the following we observe simple possibilities of practical torque estimation in induction motor drives. III.
TORQUE ESTIMATION
In the following chapter four simple torque estimation methods are discussed. The first estimate named T1 is based on the airgap torque, T2 is based on conventional motor loss components, T3 is based on loss components obtained from the converter no load run and in estimate T4 air gap torque is adjusted with slip.. A. Torque Estimation With Known Stator Resistance (T1) The stator resistance can be easily measured with a multimeter in the case of small machines, with a more sophisticated 4-wire measurement device from motor terminals, or by using a known DC-voltage and measuring the current value it causes in the windings. The resistance value can be corrected according to the winding temperature to obtain a more accurate value. The measured winding temperature or a thermalmodel-estimated value can be used to protect the motor from overheating in present-day commercial converters and it is a commonly available quantity. The stator resistance is a mandatory parameter to estimate the torque produced by the motor. With an accurate stator resistance value the air-gap torque can be calculated with a reasonable accuracy for many applications as will be shown later on. The air-gap torque equation (1) given value does not include the negative effect of rotor losses, additional losses and mechanical losses on the real torque value and thus the value given by (1) should be always greater than the real torque value. The torque estimate can, however, be so easily calculated in the discrete form of the equations (1) and (2) that it is tempting to try to find a simple way of utilizing these equations in the real torque assessment. B. Torque Estimation Based on Loss Components (T2) If a no-load test can be performed and the number of rotor bars is known, we can estimate the torque produced by the motor using the loss components.
If the number of the motor rotor bars is known or some identification protocol is used to obtain it, the stator current spectrum can be used to obtain the slip value of the motor and the rotor losses can be taken into account to the power value. The slip frequency estimation algorithms are not a subject of this paper and can be found e.g. in [7], [8]. The slip effects are clearly visible in Fig. 1. The slip =1−
( r + s) s r
(4)
,
where r, is the rotor principal slot harmonic frequency, s is the supply frequency and r is the number of rotor bars. In Fig. 1., r = [310.5, 315.8, 319.5, 322.4] Hz and using (4) with s = 25Hz, r = 28 and = 2 results in slip values of 0.041, 0.026, 0.015 and 0.007 with load values 100%, 75%, 50% and 25% respectively. With the no-load test, the friction and windage losses and iron losses of the machine can be determined and the loss components can be derived. A new torque estimate is then est
=
(
−
s
−
− Wr r
Fe
−
fw )
,
(6)
where is electric input power, s = 3 s , Wr is the rotor angular velocity, and Fe and fw are the iron losses and friction and windage losses from the no-load test. The rotor joule r losses are r
=(
−
s
−
) .
(7)
To get an adequate accuracy with this estimate in a broad operation range, multiple excitation frequencies have to be used to perform the no-load tests. C. Torque Estimation Based on Loss Components Obtained from Frequency Converter No-Load Run (T3) Since in most cases the no-load voltage curves with different excitation frequencies would be too troublesome, it is more convenient to simply drive the motor with a converter using the desired speed values. As it is seen in equation (6) we are actually interested in the constant loss part C = Fe + fw and not in the
When the motor is only driven with different speeds without altering the voltage levels we cannot segregate the iron loss components from the results to be used in determination of the rotor Joule losses using equation (7). However, the input power value magnitude is three decades as big as the iron loss value ( 1 ≫ Fe ) and neglecting the iron loss value from equation (7) does not generate a big error. Now, we can re-write equation (6) in the following form
est
−
=
s
−
where ∗ r
=(
∗ r r
−
C,PWM
(8)
,
− s) .
(9)
D. Air-Gap Torque Times Slip (T4) If the slip is obtained by some means, we can try to improve the torque estimate obtained earlier with (1) with an assumption that there exists a virtual torque component D ∗ = air-gap that is lost from the air-gap torque value given by (1) to compensate for the losses of the machine est
=
air-gap
−D
∗
»
air-gap (1
− ).
(10)
This assumption is, naturally, not physically correct since, this virtual torque component never forms in the motor and it is just introduced to correct the errors in the model used for torque estimation. In reality, the torque is not lost in a rotor but the rotor losses will decrease the rotational speed of the induction machine. IV. EXPERIMENTAL TESTS Measurements were performed on a standard 15 kW squirrel cage induction motor with a frequency converter supply and
Mechanical losses (W)
individual components when inspecting the torque estimation accuracy. Here a no-load run with four different speeds was performed and the obtained active power and constant loss power part are shown in Fig 4. Now, the measured power value includes also the additional loss component from PWM supply that should be in favour and improve the torque estimation accuracy over the estimate T2.
Fig 2. Mechanical losses obtained with a retardation test and with no-load tests using four different excitation results. The linear fit illustrates the mechanical loss estimation results if only 50 Hz grid frequency can be used to perform the no-load test.
with a sinusoidal generator supply. The motor nameplate values are shown in Table I. The load torque values applied were 25%, 50%, 75% and 100% of the motor rated torque and the fundamental voltage frequencies used in the measurements were 25%, 50%, 75% and 100% of the motor rated frequency forming a torque-speed plane of 16 points. The sinusoidal generator supply fundamental voltage and frequency were tuned to obtain similar values with both supplies. The rated torque at 25% frequency could not be driven to thermal equilibrium due to the rapid temperature rise in this operating point and a possibility to thermal breakdown of the test motor. Therefore, this measurement was terminated. The full details of the measurement setup are given in [9]. The four different supply frequencies mentioned above were used to perform the no-load test and to obtain the iron losses and friction and windage losses in these points. A resistancetemperature curve was obtained by heating the motor above the normal operating temperature and then letting the machine cool down to room temperature at standstill while continuously measuring the winding temperatures and the stator resistance. The mechanical loss results with different excitation frequencies are compared to the values obtained with the retardation test values in Fig. 2. The mechanical loss value is extracted from the no load points using the points that show no significant saturation effect and developing the curve of constant losses during the no load
TABLE I TEST MOTOR NAMEPLATE VALUES FOR 400 V, 50 HZ WITH ΔValue
Power,
15 kW
Current,
27.8 A
Voltage,
400 V
Frequency,
50 Hz
Speed,
1474 min
Power factor, cos
0.84
Efficiency,
92.7% (IE3)
Torque,
97.2 Nm
Voltage (V)
CONNECTION
Variable
Fig. 3. The iron loss curves with excitation frequencies of 12.5 Hz, 25 Hz, 37.5 Hz and 50 Hz. The loss values at each frequency has been marked with star.
TABLE II THE PARAMETERS NEEDED IN ADDITION TO POWER ANALYZER QUANTITIES FOR TORQUE ESTIMATES CALCULATION . Estimate T1 T2 T3 T4 Fig 4. The frequency converter no-load run with four different frequencies, 12.5 Hz, 25 Hz, 37.5 Hz and 50 Hz to determine the torque estimate T3. The subscript ‘electric’ represents the measured active power and ‘P,PWM’ the constant loss part from which the stator resistive losses has been removed.
test against no-load voltage squared and extrapolating a straight line to zero voltage. The intercept with zero voltage axis is the mechanical losses value. However, this method gives a negative value for mechanical losses from 12.5 Hz no-load test and the 50 Hz no-load test mechanical loss value is around 10 Watts away from the retardation test curve. From Fig. 2 we can conclude that the no-load tests are not as reliable method as the retardation test to determine the mechanical losses. It is obvious that the electric power measurement accuracy is decreased due to very low power factor during the no-load tests with induction motor. However, the discrepancy seen in the friction and windage losses has only a marginal effect on the estimated torque. It should be noted that in this case a fairly good estimate for the mechanical losses can be obtained with a simple linear line approximation from zero speed to nominal speed as shown in Fig. 2. Like the mechanical loss values, the iron loss values were determined from the data of four no-load tests with excitation frequencies of 12.5 Hz, 25 Hz, 37.5 Hz and 50 Hz. The iron loss curves are shown in Fig. 3 with the pinpointed values at each frequency that takes the resistive voltage drop in the primary winding into account. These iron loss values were used to obtain the estimate T2 for generator supply. To obtain the torque estimate T3 for frequency converter supply, a no load run with four different speeds was performed and data collected. The total run time was slightly over 2 minutes and the whole results of this run are shown in Fig. 4. The average of 15 data points marked in Fig. 4 was used to calculate the constant power value to determine the torque estimate for each speed. V. TORQUE CALCULATION FROM MEASURED VALUES The torque estimate value can be calculated from the measured time domain values after using the Clarke transformation and transfer the three phase quantities to the twoaxis domain α β
1 ⎡1 − 2 2 = ⎢ 3⎢ √3 ⎣0 2
1 ⎤ 2⎥ √3⎥ − ⎦ 2 −
U V
W
,
(8)
Additional parameters ( )
( ). (slip) ( ).
(slip)
Fe (
) and
C,PWM (
Mech. (
) from no load test(s).
) from converter no load run. (slip)
where subscript α and β correspond to two-axis domain variables and U, V and W to three phase variables. The air-gap torque at each time step (k) is in component form
3 ysα ( ) sβ ( ) − y sβ ( ) sα ( ) . (9) 2 The current and the voltage can be transformed directly to the two-axis domain using (8) and the flux can be integrated from measured voltage step-by-step air-gap (
)=
ysα ( ) = ysα(0) + [
sα (
)−
s sα (
)]∆ ,
(10)
where ∆ is the power analyzer sampling rate. The calculation is similar for the β component. Here the power analyzer sampling rate was 1 µs and the sample length was 1 second. The result obtained with (9) is the torque estimate T1 directly. The resistance value used here was based on the measured resistance, which was corrected according to winding temperature. The torque estimate values presented in the following chapter are an average value of each time step (k) over the sample length. With today’s processors’ floating point computational power, these kinds of calculations can be easily performed in real time. The parameters not available in modern power analyzers but needed for the torque estimate calculation are collected in Table II. The slip is marked in Table II because the rotor bar number is not always a known quantity and also other less complicated means than the spectrum estimation can be used to obtain the rotational speed, such as contactless sensors that are based on optics or on the detection of magnetic field disturbances. VI. RESULTS The torque estimate values’ differences compared to the measured values obtained with a state-of-the-art torque transducer using a sinusoidal generator supply are shown in Fig. 5 and using converter supply in Fig. 6. A. Generator supply The absolute difference between the first torque estimate based on the air-gap torque (T1) and the more real value is slightly increasing when the load is increased. The torque difference is from 1.8 Nm to 2.2 Nm with 12.5 Hz, from 1.8 Nm to 2.4 Nm with 25 Hz, from 2.1 Nm to 2.8 Nm 37.5 Hz and from 2.4 Nm to 3.0 Nm with 50 Hz. Thus, the difference is also increasing as a function of frequency. The average error in these 16 points is 2.3 Nm. In Fig. 5 (b) we see that the relative difference is decreased while the load is increased with all four
T1
T2
2
2
1
1
0
0
-1
-1
-2
-2
12.5 Hz
-3
25
-3 50 75 Torque (%)
100
-1
-1
-2
-2 -3
37.5 Hz 50
75
50
75
100
Torque (%) 0
25
25 Hz 25
0
-3
T4
100
50 Hz 25
Torque (%)
50
75
100
Torque (%)
Fig 5. (a) The absolute torque difference (Nm) in 16 operating points with a generator supply using three different methods. T1: ‘air-gap torque’, T2: ‘loss components from no-load tests’, T4: ‘air gap torque times (1-slip)’. T1
12.5 Hz 5
T2
T4
8
25 Hz
6 4
0
2 -5
0 -2
-10 25
50
75
100
25
Torque (%)
50
75
100
Torque (%)
10
absolute error curves that are increased with the load. The absolute difference is smallest with 25 Hz frequency from 0.2 Nm to 0.9 Nm increasing with the load. The next smallest absolute difference is with 37.5 Hz frequency (0.3 Nm to 1.0 Nm), then with 12.5 Hz frequency (0.4 to 1.1 Nm) and the greatest difference is examined with 50 Hz frequency (0.4 Nm to 1.1 Nm). The behavior of the relative difference is almost linear with all frequencies. Slight discrepancy is seen in the results, but in general the relative difference is smallest with 50% and 75% loads. The average percentage error is 1.1% in these 16 points and 0.64% when the relative difference values are scaled to the nominal load value. Here, these values were obtained with the results obtained with four different frequency no-load tests, but the difference would not have been much greater with results obtained with single no-load test as seen in Fig. 2 and Fig. 3. The third estimate, marked T4 in Fig. 5, from which the virtual torque that represents the iron and additional losses is removed, gives good results with 37.5 Hz and 50 Hz frequencies but the results are only satisfactory at 25 Hz and almost not usable at 12.5 Hz frequency. The difference at 12.5 Hz and 100% torque was 7.2 Nm and is scaled outside the figure. Equation (10) overestimates the effect on slip. Here, it is more convenient to use absolute values of relative difference, since the difference sign is changed in some measurement points. The absolute relative difference of this estimate is 3.6%, so we can say that it gives improved results over the estimate (T1) but, naturally, the expected behavior and stability of this estimate is not as good as with T1. With nominal torque value as a reference the average absolute relative difference of estimate T4 is 1.7% in these 16 points.
10
37.5 Hz
8 6
6
4
4
2
2
0
50 Hz
8
0 25
50
75
Torque (%)
100
25
50
75
100
Torque (%)
Fig 5. (b) The relative torque difference (%) in 16 operating points with generator supply using three different methods. T1: ‘air-gap torque’, T2: ‘loss components from no-load tests’, T4: ‘air gap torque times (1-slip)’.
excitation frequencies. The relative error range is 7.0 to 9.5% with 25% load decreasing to 2.2 to 3.1% with 100% load. The average percentage error is 4.6%. Sometimes it is more convenient to scale the relative error to nominal values as in the case of measurement instruments. With nominal load torque value as a reference value, the scaled average percentage error for this method with generator supply is 2.3%. The second estimate (T2) is based on the loss components calculated online and based on mechanical and iron losses that are obtained from no-load sinusoidal supply tests. The friction and windage and iron losses can be stored in a look-up table or tables while the stator and rotor resistive loss calculation can be performed online using this information. It is understandable that this kind estimate performs well with sinusoidal supply, since only the additional load loss component is missing from the total machine losses. This can be easily seen from the
B. Converter supply Fig. 6 shows the estimation results with a frequency converter supply using exactly similar procedure as with the sinusoidal generator supply. The raw numerical data with no filtering is used to calculate the estimates. In general, the estimate T1 results with converter supply are very similar to the results with sinusoidal supply. The estimation error is increasing with load and frequency and the torque difference ranges are very similar to generator supply. From 1.6 Nm to 2.5 Nm with 12.5 Hz, from 1.8 Nm to 2.6 Nm with 25 Hz, from 1.9 Nm to 2.8 Nm 37.5 Hz and from 2.2 Nm to 2.9 Nm with 50 Hz. With the generator supply, the estimate T2 performs better than T1 in all points but with converter supply the situation is reversed with 12.5 Hz frequency. The average error in these 16 points is 2.2 Nm while it was 2.3 Nm with the generator supply. From these results we can conclude that the air-gap torque estimation accuracy is similar both with generator and frequency converter supply. The torque estimate T2 based on loss components obtained from the no-load tests was expected to perform worse with converter supply than with generator supply. With converter supply we have two loss components missing from the total losses of the machine – the additional load losses and the harmonic losses generated by PWM supply. This can be confirmed by comparing the T2 results from Figs 5 (a) and 6 (a).
T1
T2
T3
2
2
1
1
0
0
-1
-1
-2 -3
T2 Generator
T4
25
-3 50 75 Torque (%)
-1
-2
50
75
-3 100
25
50
75
100
Torque (%)
Fig 6. (a) The absolute torque difference (Nm) in 16 operating points with a converter supply using four different methods. T1: ‘air-gap torque’, T2: ‘loss components from no-load tests’, T3: ‘loss components based on converter noload run’, T4: air gap torque times (1-slip). 12.5 Hz 5
T2
T3
T4
8
25 Hz
6 4
0
2 -5
0 -2
8
100
25
50
75
100
Torque (%)
37.5 Hz
8
6
6
4
4
2
2
0
50 Hz
0 25
50
75
Torque (%)
100
80
100
25
50
75
100
Torque (%)
Fig 6. (b) The relative torque difference (%) in 16 operating points with a converter supply using four different methods. T1: ‘air-gap torque’, T2: ‘loss components from no-load tests’, T3: ‘loss components based on converter noload run’, T4: ‘air gap torque times (1-slip)’.
The absolute torque estimate difference between the generator and the converter supply is increasing as a function of load but decreasing as a function of frequency. From this we can conclude that the frequency converter supply is generating a load dependent loss component. The average error with T2 is only 0.6 Nm with generator supply and 1.7 Nm with converter supply. The 16 points’ average percentage error is 3.2% and 1.7% when scaled with the nominal torque value. The torque estimate T3 is using the information obtained from the converter no-load run with four different speeds. This can be considered as an improvement over estimate T2 since now the estimate includes the harmonic losses generated by the converter PWM waveform. This fact is easily visible in Fig. 6. The estimate T3 level has changed towards zero error and the ranges changes from 0.2 Nm to 0.7 Nm, from 0.3 Nm to 1.0 Nm,
25
50
75
100
Torque (%)
Torque (%) 0
0
-0.5
-0.5
-1
25
Torque (%)
T1
60
37.5 Hz
-3 75
25 Hz
40
-1
50 Hz
50 75 Torque (%)
20
100
-2
37.5 Hz
25
-1
Torque (%)
-1
-10
-0.5
12.5 Hz
25
0
50
-0.5
25 Hz
100
0
25
0
-1
-2
12.5 Hz
T3 Converter
0
50 Hz 50
75
Torque (%)
100
25
50
75
100
Torque (%)
Fig 7. The absolute torque difference (Nm) with torque estimate T2 using generator supply and T3 with a converter supply. T2: ‘loss components from no-load tests’, T3: ‘loss components based on converter no-load run’.
0.3 Nm to 1.1 Nm and 0.3 Nm to 1.2 Nm with frequencies of 12.5 Hz, 25 Hz, 37.5 Hz and 50 Hz, respectively. The overall difference between the estimated and real value is 0.6 Nm that is comparable to T2 estimate accuracy with generator supply. The relative difference stays below one percent with 12.5 Hz frequency, is around one percent with 25 Hz frequency, and slightly over one percent with 37.5 Hz and 50 Hz frequencies that results to overall relative difference of 1.0% that corresponds to 0.6% relative difference when the nominal torque is used as a scaling value. The estimate T2 values with converter supply from Fig. 5 (a) and estimate T3 values from Fig. 6 (a) are collected in Fig. 7. From Fig. 7 we can conclude that the torque estimate T3 gives for the converter supplied motor as accurate results as the torque estimate T2 gives for the generator supplied motor. The discrepancy seen at 12.5 Hz with 100% load torque is from the fact the thermal equilibrium could not be reached in this point as discussed earlier. As is the case with other torque estimates, also estimate T4 which is T1´(1-s) performs similarly with converter supply as with generator supply since the estimate is based on T1. When comparing Figs 5 and 6, it is examined that overall all the estimates behave similarly with both supplies. This is obvious since the torque producing fundamental wave voltage – and thus the resulting flux – are similar with both supplies. VII. DISCUSSION The accuracy of the results showed in the paper depends on the particular machine design and additional load loss and harmonic loss proportion of the total losses. Here, the increased converter input voltage of 456 V RMS was used to obtain full fundamental wave voltage to motor up to the 50 Hz point. With typical 400 V RMS input voltage the motor is driven in the 50 Hz point either using field-weakening or overmodulation, which can be expected to reduce the accuracy of these estimation methods.
It is shown that the speed can be detected from the current spectrum, but naturally, the speed can also be measured by other means if necessary or more easily arranged. The methods used here do not utilize equivalent circuit parameters and the computational burden is low. These methods are suitable to be programmed in real time systems, for example in a programmable power analyzer. With the generator supply, the loss components based method (T2) can be improved by using a fixed value for additional load losses that has not been taken into account. The fixed value given in IEC standard [10] is 1.9% from electric input power of 15 kW machine. In [11], it is shown that the additional load losses are 0.5% to 1.5% from the input electric power that would almost perfectly match the average torque difference of 1.7% scaled with a nominal torque value. Either of these values would improve the absolute estimation accuracy with generator supply. VIII. CONCLUSION Torque can be estimated with a reasonable accuracy using the motor terminal quantities measured with a power analyzer, while the torque measurement from the shaft is not always necessarily possible. Simple algorithms can used to improve the estimation accuracy. The constant loss term is easily estimated from the converter no-load run and this kind of loss component based estimate performs very well according to the results presented here. A simple method to improve the induction motor torque estimation accuracy from the air-gap torque is presented. By performing the no-load tests or converter no-load run before the actual measurement, the torque estimation accuracy can give relatively good results over the normal operating points from 25…100% speed and 25…100% load without increasing the computational burden much.
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